Nonparametric Regression (ELMR Chapter 14)

Biostat 200C

Display system information and load tidyverse and faraway packages

sessionInfo()

## R version 3.6.2 (2019-12-12)
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library(tidyverse)

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library(faraway)

Parametric vs nonparametric models

• Given a regressor/predictor \(x_1,\ldots,x_n\) and response \(y_1, \ldots, y_n\), where \[ y_i = f(x_i) + \epsilon_i, \] where \(\epsilon_i\) are iid with mean zero and unknown variance \(\sigma^2\).

• Parametric approach assumes that \(f(x)\) belongs to a parametric family \(f(x \mid \boldsymbol{\beta})\). Examples are \[\begin{eqnarray*} f(x \mid \boldsymbol{\beta}) &=& \beta_0 + \beta_1 x \\ f(x \mid \boldsymbol{\beta}) &=& \beta_0 + \beta_1 x + \beta_2 x^2 \\ f(x \mid \boldsymbol{\beta}) &=& \beta_0 + \beta_1 x^{\beta_2}. \end{eqnarray*}\]

• Nonparametric approach assumes \(f\) is from some smooth family of functions.

• Three datasets.

  • exa: \(f(x) = \sin^3 (2 \pi x^3)\).

  exa <- as_tibble(exa) %>% print()

  ## # A tibble: 256 x 3
  ##          x       y     m
  ##      <dbl>   <dbl> <dbl>
  ##  1 0.00480 -0.0339     0
  ##  2 0.0086   0.165      0
  ##  3 0.0117   0.0245     0
  ##  4 0.017    0.178      0
  ##  5 0.0261  -0.347      0
  ##  6 0.0299  -0.755      0
  ##  7 0.0307   0.355      0
  ##  8 0.0315   0.0401     0
  ##  9 0.0331   0.106      0
  ## 10 0.034    0.122      0
  ## # … with 246 more rows

  exa %>%
    ggplot(mapping = aes(x = x, y = y)) + 
    geom_point() +
    geom_smooth(span = 0.2) + # small span give more wiggleness
    geom_line(mapping = aes(x = x, y = m))

  ## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

  

  • exb: \(f(x) = 0\).

  exb <- as_tibble(exb) %>% print()

  ## # A tibble: 50 x 3
  ##        x       y     m
  ##    <dbl>   <dbl> <dbl>
  ##  1  0.02  0.781      0
  ##  2  0.04  1.64       0
  ##  3  0.06 -0.363      0
  ##  4  0.08 -0.0540     0
  ##  5  0.1  -0.804      0
  ##  6  0.12  0.907      0
  ##  7  0.14 -0.843      0
  ##  8  0.16 -0.229      0
  ##  9  0.18 -0.499      0
  ## 10  0.2   1.20       0
  ## # … with 40 more rows

  exb %>%
    ggplot(mapping = aes(x = x, y = y)) + 
    geom_point() +
    geom_smooth() +
    geom_line(mapping = aes(x = x, y = m))

  ## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

  

  • faithful: data on Old Faithful geyser in Yellowstone National Park.

  faithful <- as_tibble(faithful) %>% print()

  ## # A tibble: 272 x 2
  ##    eruptions waiting
  ##        <dbl>   <dbl>
  ##  1      3.6       79
  ##  2      1.8       54
  ##  3      3.33      74
  ##  4      2.28      62
  ##  5      4.53      85
  ##  6      2.88      55
  ##  7      4.7       88
  ##  8      3.6       85
  ##  9      1.95      51
  ## 10      4.35      85
  ## # … with 262 more rows

  faithful %>%
    ggplot(mapping = aes(x = eruptions, y = waiting)) + 
    geom_point() +
    geom_smooth() # small span give more wiggleness

  ## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

  

Kernel estimators

• Moving average estimator \[ \hat f_\lambda(x) = \frac{1}{n\lambda} \sum_{j=1}^n K\left( \frac{x-x_j}{\lambda} \right) Y_j = \frac{1}{n} \sum_{j=1}^n w_j Y_j, \] where \[ w_j = \lambda^{-1} K\left( \frac{x-x_j}{\lambda} \right), \] and \(K\) is a kernel such that \(\int K(x) \, dx = 1\). \(\lambda\) is called the bandwidth, window width or smoothing parameter.

• When \(x\)s are spaced unevenly, the kernel estimator can give poor results. This is improved by the Nadaraya-Watson estimator \[ f_\lambda(x) = \frac{\sum_{j=1}^n w_j Y_j}{\sum_{j=1}^n w_j}. \]

• Asymptotics of kernel estimators \[ \text{MSE}(x) = \mathbb{E} [f(x) - \hat f_\lambda(x)]^2 = O(n^{-4/5}). \] Typical parametric estimator has \(\text{MSE}(x) = O(n^{-1})\) if the parametric model is correct.

• Choice of kernel. Ideal kernel is smooth, compact, and amenable to rapid computation. The optimal choice under some standard assumptions is the Epanechnikov kernel \[ K(x) = \begin{cases} \frac 34 (1 - x^2) & |x| < 1 \\ 0 & \text{otherwise} \end{cases}. \]

• Choice of smoothing parameter \(\lambda\). Small \(\lambda\) gives more wiggly curves, while large \(\lambda\) yields smoother curves.

for (bw in c(0.1, 0.5, 2)) {
  with(faithful, {
    plot(waiting ~ eruptions, col = gray(0.75))
    lines(ksmooth(eruptions, waiting, "normal", bw))
  })
}

• Cross-valiation (CV) choose the \(\lambda\) that minimizes the criterion \[ \text{CV}(\lambda) = \frac 1n \sum_{j=1}^n [y_j - \hat f_{\lambda(j)}(x_j)]^2, \] where \((j)\) indicates that point \(j\) is left out of the fit.

library(sm)

## Package 'sm', version 2.2-5.6: type help(sm) for summary information

with(faithful, 
     sm.regression(eruptions, waiting, h = h.select(eruptions, waiting)))

with(exa, sm.regression(x, y, h = h.select(x, y)))

with(exb, sm.regression(x, y, h = h.select(x, y)))

Splines

Smoothing splines

• We choose \(\hat f\) to minize the modified least squares criterion \[ \frac 1n \sum_i [y_i - f(x_i)]^2 + \lambda \int [f''(x)]^2 \, dx, \] where \(\lambda > 0\) is the smoothing paramter and \(\int [f''(x)]^2 \, dx\) is a roughness penalty. For large \(\lambda\), the minimizer \(\hat f\) is smoother; for smaller \(\lambda\), the minizer \(\hat f\) is rougher. This is the smoothing spline fit.

• The minimizer takes a special form: \(\hat f\) is a cubic spline (piecewise cubic polynomial in each interval \((x_i, x_{i+1})\)).

• The tuning parameter \(\lambda\) is chosen by cross-validation in R.

with(faithful, {
  plot(waiting ~ eruptions, col = gray(0.75))
  lines(smooth.spline(eruptions, waiting), lty = 2)
})

with(exa, {
  plot(y ~ x, col = gray(0.75))
  lines(x, m) # true model
  lines(smooth.spline(x, y), lty = 2)
})

with(exb, {
  plot(y ~ x, col = gray(0.75))
  lines(x, m) # true model
  lines(smooth.spline(x, y), lty = 2)
})

Regression splines

• The regresison spline fit differs from the smoothing splines in that the number of knots can be much smaller than the sample size.

• Piecewise linear splines:

rhs <- function(x, c) ifelse(x > c, x - c, 0)
curve(rhs(x, 0.5), 0, 1)

• Define some knots for Example A

(knots <- 0:9 / 10)

##  [1] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

and compute a design matrix of splines with knots at these points for each \(x\):

dm <- outer(exa$x, knots, rhs)
matplot(exa$x, dm, type = "l", col = 1, xlab = "x", ylab="")

• Compute and dipslay the regression spline fit.

lmod <- lm(exa$y ~ dm)
plot(y ~ x, exa, col = gray(0.75))
lines(exa$x, predict(lmod))

• We can acheive better fit by using more knots in denser regions of greater curvature.

newknots <- c(0, 0.5, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95)
dmn <- outer(exa$x, newknots, rhs)
lmod <- lm(exa$y ~ dmn)
plot(y ~ x, data = exa, col = gray(0.75))
lines(exa$x, predict(lmod))

• High-order splines can produce a smoother fit. The bs() function can be used to generate the appropriate spline basis. The default is cubic B-splines.

library(splines)

matplot(bs(seq(0, 1, length = 1000), df = 12), type = "l", ylab="", col = 1)

lmod <- lm(y ~ bs(x, 12), exa)
plot(y ~ x, exa, col = gray(0.75))
lines(m ~ x, exa)
lines(predict(lmod) ~ x, exa, lty = 2)

Local polynomials

• Both kernel and spline smoothers are vulnerable to outliers.

• Local polynomial method fit a polynomial in a window using robust methods, then the predicted response at the middle of the window is the fitted value. This procedure is repeated by sliding the window over the range of the data. lowess or loess functions in R.

• We need to choose the order of polynomial and window width. Default window width is 3/4 of data. Default polynomial order is 1 (linear).

• Examples.

with(faithful, {
  plot(waiting ~ eruptions, col = gray(0.75))
  f <- loess(waiting ~ eruptions)
  i <- order(eruptions)
  lines(f$x[i], f$fitted[i])
})

with(exa, {
  plot(y ~ x, col = gray(0.75))
  lines(m ~ x)
  f <- loess(y ~ x)
  lines(f$x, f$fitted, lty = 2)
  # try smaller span (proportion of the range)
  f <- loess(y ~ x, span = 0.22)
  lines(f$x, f$fitted, lty = 5)
})

with(exb, {
  plot(y ~ x, col = gray(0.75))
  lines(m ~ x)
  f <- loess(y ~ x)
  lines(f$x, f$fitted, lty = 2)
  # splan = 1 means whole span of data (smoothest)
  f <- loess(y ~ x, span = 1)
  lines(f$x, f$fitted, lty = 5)
})

• Pointwise confidence band is obtained by the local parametric fit for smoothing splines and loess.

ggplot(data = exa, mapping = aes(x = x, y = y)) +
  geom_point(alpha = 0.25) + 
  geom_smooth(method = "loess", span = 0.22) + 
  geom_line(mapping = aes(x = x, y = m), linetype = 2)

## `geom_smooth()` using formula 'y ~ x'

• Simultaneous confidence band can be constructed by the mgcv package.

library(mgcv)

## Loading required package: nlme

## 
## Attaching package: 'nlme'

## The following object is masked from 'package:dplyr':
## 
##     collapse

## This is mgcv 1.8-31. For overview type 'help("mgcv-package")'.

ggplot(data = exa, mapping = aes(x = x, y = y)) +
  geom_point(alpha = 0.25) + 
  geom_smooth(method = "gam", span = 0.22, formula = y ~ s(x, k = 20)) + 
  geom_line(mapping = aes(x = x, y = m), linetype = 2)

Wavelets

• In general, we approximate a curve by a family of basis functions \[ \hat f(x) = \sum_i c_i \phi_i(x), \] where the basis functions \(\phi_i\) are given and the coefficients \(c_i\) are estimated.

• Ideally we would like the basis functions \(\phi_i\) to be (1) compactly supported (adatped to local data points) and (2) orthogonal (fast computing).

• Orthogonal polynomials and Fourier basis are not compactly supported.

• Cubic B-splines are compactly supported but not orthogonal.

• Wavelets are both compactly supported and orthogonal.

• Haar basis. The mother wavelet function on [0, 1] \[ w(x) = \begin{cases} 1 & x \le 1/2 \\ -1 & x > 1/2 \end{cases}. \] Next two members are defined on [0, 1/2) and [1/2, 1) by rescaling the mother wavelet to these two intervals. In general, at level \(j\) \[ h_n(x) = 2^{j/2} w(2^j x - k), \] where \(n = 2^j + k\) and \(0 \le k \le 2^j\).

library(wavethresh)

## Loading required package: MASS

## 
## Attaching package: 'MASS'

## The following object is masked from 'package:sm':
## 
##     muscle

## The following object is masked from 'package:dplyr':
## 
##     select

## WaveThresh: R wavelet software, release 4.6.8, installed

## Copyright Guy Nason and others 1993-2016

## Note: nlevels has been renamed to nlevelsWT

wds <- wd(exa$y, filter.number = 1, family = "DaubExPhase")
draw(wds)

plot(wds)

## [1] 6.12868 6.12868 6.12868 6.12868 6.12868 6.12868 6.12868 6.12868

Let’s only retain 3 levels of coefficients.

wtd <- threshold(wds, policy = "manual", value = 9999)
fd <- wr(wtd)
plot(y ~ x, exa, col = gray(0.75))
lines(m ~ x, exa)
lines(fd ~ x, exa, lty = 5, lwd = 2)

Or we can zero out only the small coefficients.

wtd2 <- threshold(wds)
fd2 <- wr(wtd2)
plot(y ~ x, exa, col = gray(0.75))
lines(m ~ x, exa)
lines(fd2 ~ x, exa, lty = 5, lwd = 2)

• We may perfer to use continuous wavelet basis functions.

wds <- wd(exa$y, filter.number = 2, bc = "interval")
draw(filter.number = 2, family = "DaubExPhase")

plot(wds)

## [1] 3.3061 3.3061 3.3061 3.3061 3.3061 3.3061 3.3061 3.3061

Now we zero out small coefficients.

wtd <- threshold(wds)
fd <- wr(wtd)

## Warning in wr.wd(wtd): All optional arguments ignored for "wavelets on the
## interval" transform

plot(y ~ x, exa, col = gray(0.75))
lines(m ~ x, exa)
lines(fd ~ x, exa, lty=2)

• For the Old Faithful data.

x <- with(faithful, (eruptions - min(eruptions)) / (max(eruptions) - min(eruptions)))
gridof <- makegrid(x, faithful$waiting)
wdof <- irregwd(gridof, bc="symmetric")
wtof <- threshold(wdof)
wrof <- wr(wtof)
plot(waiting ~ eruptions, faithful, col = grey(0.75))
with(faithful, lines(seq(min(eruptions), max(eruptions), len=512), wrof))